Phase behavior of a model bilayer membrane with coupled leaves

Abstract

We discuss the thermodynamic behavior of a bilayer composed of two coupled leaves and derive the Gibbs Phase Rule for such a system. A simple phenomenological model of such a system is considered in which the state of the bilayer is specified by the relative number of ordering lipids in the outer leaf, and in the inner leaf. Two cases are treated. In the first, both inner and outer leaves could undergo phase separation when uncoupled from one another. The bilayer can exist in four different phases, and can exhibit three-phase coexistence. In the second case, an outer layer which can undergo phase separation by itself is coupled to an inner leaf which cannot. We find that when the coupling is weak, the bilayer can exist in only two phases, one in which the outer layer is rich in ordering lipids and the inner leaf is somewhat richer in them than when uncoupled, and another in which the outer layer is poor in ordering lipids and the inner leaf is poorer in them than when uncoupled. Increasing the coupling increases the effect on the inner leaf composition due to small changes in those of the outer leaf. For sufficiently large coupling, a phase transition occurs and the bilayer exhibits four phases as in the first case considered. Our results are in accord with several observations made recently.

Figure 1

(a) Phase diagram in the x,y plane of the bilayer for r = 1, so that the leaves are essentially identical, and a weak coupling β = 0.5. The order parameter, y, of the outer leaf is shown in units of $\widehat{Y},$ and that of the inner leaf, x, is shown in units of $\widehat{X}=r\widehat{Y}\text{.}$ The tie lines end on the binodals of the coupled system shown with solid lines. The dashed-dotted lines denote the binodals of the uncoupled system. There are four possible phases of the bilayer; one in which the order parameters in both leaves are positive, (R,R′), one in which they are both negative, (P,P′), and two phases in which the order parameter in one leaf is positive while that in the other leaf is negative. Note that the region of four-phase coexistence in the uncoupled system breaks into two regions of three-phase coexistence connected by a region of two-phase coexistence between (R,R′) and (P,P′). (b) Phase diagram in the x,y plane of the bilayer for r = 1, and a stronger coupling β = 3.0. The point AA represents the state of a bilayer which consists of coupled, identical, leaves such that the system is in the one-phase region (P,P′); BB represents a bilayer of coupled, identical, leaves of a different composition such that the system is well within the coexistence region between (R,R′) and (P,P′), while the bilayer CC is barely within this two-phase region. The point AB represents the result of making a bilayer with one leaf of A and the other of B. It is within the coexistence region. The point AC represents the result of making a bilayer with one leaf of A and the other of C. It is within the one-phase region (P,P′).

Figure 2

Phase diagrams for a system at a temperature at which the outer leaf can undergo phase separation when uncoupled from the inner leaf, but the inner leaf cannot undergo a phase separation when uncoupled from the outer leaf. The value of r = 1. (a) Phase diagram of the uncoupled system, i.e., β = 0. Solid lines and dashed lines denote the binodals and spinodals, respectively. (b) Same but for β = 0.75 and (c) for β = 2.25. For all three values of the coupling, the bilayer can exist in only two phases; that in which the inner leaf is disordered while the outer leaf is rich in ordering lipids (d,R′), and that in which the inner leaf is disordered while the outer leaf is poor in ordering lipids, (d,P′).

Figure 3

Phase diagrams for a system at a temperature at which the outer leaf can undergo phase-separation when uncoupled from the inner leaf, but the inner leaf cannot undergo a phase separation when uncoupled from the outer leaf. The value of r = 1, and the coupling is β = 4.0. (a) Phase diagram in the x,y plane. There is now an additional coexistence region between one phase in which the outer leaf is rich in saturated lipids as is the inner leaf, (R,R′), and another in which the outer leaf is rich is such lipids and the inner leaf is poor in them, (P,R′). (b) Upper-left quadrant of the phase diagram in the chemical potential plane. The chemical potentials μ_i and μ_o are given in units of ${\widehat{X}}^3$ and ${\widehat{Y}}^3,$ respectively. The lower-right quadrant follows by symmetry, and the other two display no phase boundaries.